Pregroup representable expansions of residuated lattices

2026-01-22

AI Summary

This study investigates relational representations of distributive involutive FL-algebras (DInFL-algebras) in the non-Boolean lattice setting. By introducing pregroup structures and integrating residuated lattices with order-reversing operations, the authors construct, for the first time, a relational model for DInFL-algebras admitting non-Boolean lattice reducts. Furthermore, under the condition that the pregroup is equipped with a specific order-reversing unary operation, this representation framework is extended to distributive quasi-relational algebras. The work not only achieves an effective relational representation of DInFL-algebras over finite pregroups but also provides novel universal algebraic tools and representational pathways for broader algebraic semantics.

Technology Category

Knowledge Representation and Reasoning: Description LogicsReasoning under Uncertainty: Relational Probabilistic ModelsMachine Learning: Statistical Relational/Logic Learning

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSearch and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Group representable relation algebras play an important role in the study of representable relation algebras. The class of distributive involutive FL-algebras (DInFL-algebras) generalises relation algebras, as well as Sugihara monoids and MV-algebras. We construct DInFL-algebras from pregroups and show that they can be represented as algebras of binary relations. Even for finite pregroups we obtain relational representations of DInFL-algebras with non-Boolean lattice reducts. If the pregroup is enriched with a particular unary order-reversing operation, then our construction yields representation results for distributive quasi relation algebras.

Problem

• pregroup
• residuated lattice
• relation algebra
• DInFL-algebra
• relational representation

Innovation

• pregroup
• DInFL-algebra
• relational representation
• residuated lattice
• quasi relation algebra